geometric representation theory
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Equivariant homotopy theory is homotopy theory for the case that a group $G$ acts on all the topological spaces or other objects involved, hence the homotopy theory of topological G-spaces.
The canonical homomorphisms of topological $G$-spaces are $G$-equivariant continuous functions, and the canonical choice of homotopies between these are $G$-equivariant continuous homotopies (for trivial $G$-action on the interval). A $G$-equivariant version of the Whitehead theorem says that on G-CW complexes these $G$-equivariant homotopy equivalences are equivalently those maps that induce weak homotopy equivalences on all fixed point spaces for all subgroups of $G$ (compact subgroups, if $G$ is allowed to be a Lie group). By Elmendorf's theorem, this, in turn, is equivalent to the (∞,1)-presheaves over the orbit category of $G$. See below at In topological spaces – Homotopy theory.
(Beware that $G$-homotopy theory is crucially different from (namely “finer” and “more geometric” than) the homotopy theory of the ∞-actions of the underlying homotopy type ∞-group of $G$, and this is so even when $G$ is a discrete group, see below).
The union of $G$-equivariant homotopy theories as $G$ is allowed to vary is global equivariant homotopy theory.
The direct stabilization of equivariant homotopy theory is the theory of spectra with G-action. More generally there is a concept of G-spectra and they are the subject of equivariant stable homotopy theory.
The concept of cohomology of equivariant homotopy theory is equivariant cohomology:
cohomology in the presence of ∞-group $G$ ∞-action:
Borel equivariant cohomology | $\leftarrow$ | general (Bredon) equivariant cohomology | $\rightarrow$ | non-equivariant cohomology with homotopy fixed point coefficients |
---|---|---|---|---|
$\mathbf{H}(X_G, A)$ | trivial action on coefficients $A$ | $[X,A]^G$ | trivial action on domain space $X$ | $\mathbf{H}(X, A^G)$ |
Let $G$ be a discrete group.
A topological G-space is a topological space equipped with a $G$-action.
Let $I = \mathbb{R}$ be the interval object $({*} \stackrel{0}{\to} I \stackrel{1}{\leftarrow} {*})$ regarded as a $G$-space by equipping it with the trivial $G$-action.
A $G$-homotopy $\eta$ between $G$-maps, $f, g : X \to Y$, is a left homotopy with respect to this $I$
(e.g. May 96, p. 15)
(models for $G$-equivariant spaces)
Consider the following three homotopical categories that model $G$-spaces:
Write
for the full subcategory of G-CW-complexes, regarded as equipped with the structure of a category with weak equivalences by taking the weak equivalences to be the $G$- homotopy equivalences according to def. 2.
Write
for all of $G Top$ equipped with weak equivalences given by those morphisms $(f : X \to Y) \in G Top$ that induce on for all subgroups $H \subset G$ weak homotopy equivalences $f^H : X^H \to Y^H$ on the $H$-fixed point spaces, in the standard Quillen model structure on topological spaces (i.e. inducing isomorphism on homotopy groups).
Write
for the projective global model structure on functors from the opposite category of the orbit category $O_G$ of $G$ to the Top (with its Quillen model structure on topological spaces).
The following theorem (Elmendorf's theorem) says that these models all present the same homotopy theory.
(Elmendorf’s theorem)
The homotopy categories of all three homotopical categories in def. 3 are equivalent:
where the equivalence is induced by the functor that sends a $G$-space to the presheaf that it represents.
This is stated as (May 96, theorem VI.6.3).
At topological ∞-groupoid it is discussed that the category Top of topological spaces may be understood as the localization of an (∞,1)-category $Sh_{(\infty,1)}(Top)$ of (∞,1)-sheaves on $Top$, at the collection of morphisms of the form $\{X \times I \to X\}$ with $I$ the real line.
The analogous statement is true for $G$-spaces: the equivariant homotopy category is the homotopy localization of the category of $\infty$-stacks on $G Top$.
More in detail: let $G Top$ be the site whose objects are $G$-spaces that admit $G$-equivariant open covers, morphisms are $G$-equivariant maps and morphism $Y \to X$ is in the coverage if it admits a $G$-equivariant splitting over such $G$-equivariant open covers.
Write
for the corresponding hypercomplete local model structure on simplicial sheaves.
Let $I$ be the unit interval, the standard interval object in Top, equipped with the trivial $G$-action, regarded as an object of $G Top$ and hence in $sSh(G Top)$.
Write
for the left Bousfield localization at thecollection of morphisms $\{X \stackrel{Id \times 0}{\to} X \times I\}$.
Then the homotopy category of $sSh(G Top)_{loc}^I$ is the equivariant homotopy category described above
This is (Morel-Voevodsky 03, example 3, p. 50).
The above constructions may be unified to apply “for all groups at once”, this is the content of global equivariant homotopy theory.
Let $G$ be a finite group as above. We describe the generalizaton of the above story as Top is replaced by a more general model category $C$ (Guillou).
Let $C$ be a cofibrantly generated model category with generating cofibrations $I$ and generating acyclic cofibrations $J$.
There is a cofibrantly generated model category
on the functor category from the orbit category of $G$ to $C$ by taking the generating cofibrations to be
and the generating acyclic cofibrations to be
Let $\mathbf{B}G$ be the delooping groupoid of $G$ and let
be the functor category from $\mathbf{B}G$ to $C$ – the category of objects in $C$ equipped with a $G$-action equipped with a set of generatinc (acyclic) cofibrations
and the generating acyclic cofibrations to be
This defines a cofibrantly generated model category if $[\mathbf{B}G^{op}, C]$ has a cellular fixed point functor (see…).
(generalized Elmendorf’s theorem)
There is a Quillen adjunction
and a Quillen equivalence
This is proposition 3.1.5 in Guillou.
The assumption on the model category $C$ entering the generalized Elmendorf theorem above is satisfied in particular by every left Bousfield localization
of the global projective model structure on simplicial presheaves onany small category $C$ at any set $A$ of morphisms, i.e. for every combinatorial model category $C$. This is example 4.4 in Guillou.
For $A = \{C(\{U_i\}) \to X\}$ the collection of Cech covers for all covering families of a Grothendieck topology on $D$, this are the standard models for ∞-stack (∞,1)-toposes $\mathbf{H}$.
This way the above theorem provides a model for $G$-equivariant refinements of ∞-stack (∞,1)-toposes.
For instance, in motivic homotopy theory one considers cohomology in a homotopy localization of the ∞-stack (∞,1)-topos on the Nisnevich site, presented by $C := L_{Cech} SPSh(Nis)$ . Its $G$-equivariant version as above should be the right context for the Bredon $G$-equivariant cohomology refinement of such cohomology theories, such as motivic cohomology.
This is example 4.5 in Guillou.
(Actually here one localizes moreover at hypercovers and at A1-homotopies.)
By Elmendorf's theorem the $G$-equivariant homotopy theory is an (∞,1)-topos.
By (Rezk 14) $G Top$ is also the base (∞,1)-topos of the cohesion of the global equivariant homotopy theory sliced over $\mathbf{B}G$.
The stabilization of the (∞,1)-topos $G Top \simeq PSh_\infty(Orb_G)$ is the equivariant stable homotopy theory of spectra with G-action (“naive G-spectra”).
For $G$ a discrete group (geometrically discrete) the homotopy theory of G-spaces which enters Elmendorf's theorem is different (finer) than the standard homotopy theory of $G$-∞-actions, which is presented by the Borel model structure (see there for more, and see (Guillou)).
circle group-equivariant homotopy theory may be presented by cyclic sets.
Equivariant homotopy theory is to equivariant stable homotopy theory as homotopy theory is to stable homotopy theory.
Rezk-global equivariant homotopy theory:
cohesive (∞,1)-topos | its (∞,1)-site | base (∞,1)-topos | its (∞,1)-site |
---|---|---|---|
global equivariant homotopy theory $PSh_\infty(Glo)$ | global equivariant indexing category $Glo$ | ∞Grpd $\simeq PSh_\infty(\ast)$ | point |
… sliced over terminal orbispace: $PSh_\infty(Glo)_{/\mathcal{N}}$ | $Glo_{/\mathcal{N}}$ | orbispaces $PSh_\infty(Orb)$ | global orbit category |
… sliced over $\mathbf{B}G$: $PSh_\infty(Glo)_{/\mathbf{B}G}$ | $Glo_{/\mathbf{B}G}$ | $G$-equivariant homotopy theory of G-spaces $L_{we} G Top \simeq PSh_\infty(Orb_G)$ | $G$-orbit category $Orb_{/\mathbf{B}G} = Orb_G$ |
A standard text is
Other survey includes
Stefan Schwede, appendix A.4 of of Symmetric spectra (2012)
Michael Hill, Michael Hopkins, Douglas Ravenel, section 1.3 of The Arf-Kervaire problem in algebraic topology: Sketch of the proof (pdf)
(with an eye towards application to the Arf-Kervaire invariant problem)
The generalization of the homotopy theory of $G$-spaces and of Elmendorf’s theorem to that of $G$-objects in more general model categories is in
and further discussed in
See also
Discussion in the context of global equivariant homotopy theory is in
Discussion via homotopy type theory is in